Journal of Optimization Theory and Applications

, Volume 156, Issue 1, pp 79–95

Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators

Authors

  • Michal Fec̆kan
    • Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and InformaticsComenius University
    • Mathematical InstituteSlovak Academy of Sciences
    • Department of MathematicsGuizhou University
  • Yong Zhou
    • Department of MathematicsXiangtan University
Article

DOI: 10.1007/s10957-012-0174-7

Cite this article as:
Fec̆kan, M., Wang, J. & Zhou, Y. J Optim Theory Appl (2013) 156: 79. doi:10.1007/s10957-012-0174-7

Abstract

The paper is concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach spaces. With the help of two new characteristic solution operators and their properties, such as boundedness and compactness, we present the controllability results corresponding to two admissible control sets via the well-known Schauder fixed point theorem. Finally, an example is given to illustrate our theoretical results.

Keywords

ControllabilityFractional derivativeFunctional evolution equationsSobolevCharacteristic solution operators

1 Introduction

A strong motivation for studying fractional evolution equations comes from the fact they have been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, and economics. For more details on the theory and application in this field, one can see the monographs of Diethelm [1], Kilbas et al. [2], Miller and Ross [3], Podlubny [4] and Tarasov [5].

Recently, the existence of mild solutions, Mittag–Leffer–Ulam stability, controllability and optimal controls for all kinds of fractional semilinear evolution system in Banach spaces have been reported by many researchers. We refer the reader to El-Borai [6, 7], Balachandran and Park [8], Zhou and Jiao [9, 10], Hernández et al. [11], Wang and Zhou [1214], Sakthivel et al. [15], Debbouchea and Baleanu [16], Wang et al. [1723], Kumar and Sukavanam [24], Li et al. [25] and the references therein.

It is remarkable that Balachandra and Dauer [26] provide some sufficient conditions for controllability of integer functional evolution equations of Sobolev type by virtue of the theory of semigroup theory via the techniques of fixed point theorem. Very recently, Li et al. [27] obtain the existence results for fractional evolution equations of Sobolev type by virtue of the theory of propagation family via the techniques of the measure of noncompactness and the condensing maps. However, controllability of fractional functional evolution equations of Sobolev type have not been extensively via the theory of semigroup theory or propagation family.

Motivated by [9, 26, 27], we offer to study the controllability of a class of fractional functional evolution equations of Sobolev type via the theory of semigroup theory. Our aim in this paper is to provide some suitable sufficient conditions for the controllability results corresponding to two admissible control sets without assuming the semigroup is compact. To achieve our purpose, we have to introduce two characteristic solution operators and study their properties such as boundedness and compactness. To prove the controllability results we follow the techniques similar to that of [26] with some necessary modifications so as to be compatible with fractional functional evolution equations. But we emphasize that the assumptions on the closed operator for E (see [C1] in [26]) and the compact resolvent operator (see [C4] in [26]) does not need. Here, we will remove these redundant conditions to obtain the controllability results.

2 Preliminaries

Let X and Y be two real Banach spaces. We consider the following fractional functional evolution equations of Sobolev type:
$$ \left\{ \begin{array}{l@{\quad}l} ^{C}_{0}\!D^{q}_{t}(Ex(t))+Ax(t)=f(t,x_{t})+Bu(t),&t\in J:=[0,a],\\ [1pt] x(t)=\phi(t),&-r\leq t\leq 0, \end{array} \right. $$
(1)
where \(^{C}_{0}\!D_{t}^{q}\) is the Caputo fractional derivative of order 0<q<1 with the lower limit zero, the operators A:D(A)⊂XY and E:D(E)⊂XY, the state x(⋅) takes values in X and the control function u(⋅) is given in https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq2_HTML.gif, the Banach space of admissible control functions with U a Banach space where either https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq3_HTML.gif for \(\frac{1}{2}<q<1\) or https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq5_HTML.gif for 0<q<1. Moreover, B is a bounded linear operator from U into Y. f:J×CY with C:=C([−r,0],X) will be specified later. x:J:=[−r,a]→X is continuous, xt is the element of C defined by xt(s):=x(t+s),−rs≤0. The domain D(E) of E becomes a Banach space with norm ∥xD(E):=∥ExY, xD(E) and ϕC(E):=C([−r,0],D(E)).

Let us first recall the following known definitions. For more details, see [2].

Definition 2.1

The fractional integral of the order γ with the lower limit zero for a function f:[0,∞[→ℝ is defined as
$$_{0}I_{t}^{\gamma} f(t):=\frac{1}{\varGamma(\gamma)}\int_{0}^{t}\frac{f(s)}{(t-s)^{1-\gamma}}\,ds,\quad t>0,~\gamma>0, $$
provided the right-hand be point-wise defined on [0,∞[, where Γ(⋅) is the Gamma function, which is defined by \(\varGamma(\gamma):=\int_{0}^{\infty}t^{\gamma-1}e^{-t}\,dt\).

Definition 2.2

The Riemann–Liouville derivative of the order γ with the lower limit zero for a function f:[0,∞[→ℝ can be written as
$$^L_{0}D_{t}^{\gamma} f(t):=\frac{1}{\varGamma(n-\gamma)}\frac{d^n}{dt^n}\int_{0}^{t}\frac{f(s)}{(t-s)^{\gamma+1-n}}\,ds,\quad t>0, n-1<\gamma<n. $$

Definition 2.3

The Caputo derivative of order γ for a function f:[0,∞[→ℝ can be written as
$$^C_{0}D_{t}^{\gamma}f(t):=~^L_{0}D_{t}^{\gamma}\left[f(t)-\sum^{n-1}_{k=0}\frac {t^k}{k!} f^{(k)}(0)\right], \quad t>0,~n-1<\gamma <n. $$

Remark 2.1

(i) If f(t)∈Cn[0,∞[, then
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equd_HTML.gif
for n−1<γ<n;

(ii) The Caputo derivative of a constant is equal to zero;

(iii) If f is an abstract function with values in X, then integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner’s sense.

3 Characteristic Solution Operators and their Properties

In this section, we consider the following fractional functional evolution equations:
$$ \left\{ \begin{array}{l@{\quad}l} ^{C}_{0}D^{q}_{t}(Ex(t))+Ax(t)=f(t,x_{t}),& t\in J,\\ [4pt] x(t)=\phi(t),& -r\leq t\leq 0. \end{array} \right. $$
(2)
We introduce the following assumptions on the operators A and E.
(H1):

A and E are linear operators, and A is closed.

(H2):

D(E)⊂D(A) and E is bijective.

(H3):

Linear operator E−1:YD(E)⊂X is compact (which implies that E−1 is bounded).

Note (H3) implies that E is closed since the fact: E−1 is closed and injective, then its inverse is also closed. It comes from (H1)–(H3) and the closed graph theorem, we obtain the boundedness of the linear operator −AE−1:YY. Consequently, −AE−1 generates a semigroup {T(t),t≥0}, \(T(t):=e^{-AE^{-1}t}\). We suppose M1:=supt≥0T(t)∥<∞.

According to Definitions 2.1–2.3, it is suitable to rewrite the system (2) in the equivalent fractional integral equation
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equ3_HTML.gif
(3)
provided that the integral in (3) exists.

Lemma 3.1

If the formula (3) holds, then we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Eque_HTML.gif
Herehttps://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq8_HTML.gifandhttps://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq9_HTML.gifare called characteristic solution operators and given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equ4_HTML.gif
(4)
where
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equ5_HTML.gif
(5)
ξqis a probability density function defined on ]0,∞[.

Proof

For λ>0. Formally applying the following Laplace transforms:
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equf_HTML.gif
to the first equation in (3), we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equg_HTML.gif
which implies that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equh_HTML.gif
where I is the identity operator defined on Y. Thus,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equi_HTML.gif
Now, we consider the one-sided stable probability density
$$\varpi_{q}(\theta):=\frac{1}{\pi}{\sum_{n=1}^{\infty}} (-1)^{n-1}{\theta}^{-qn-1}\frac{\varGamma(nq+1)}{n!}\sin(n\pi q),\quad\theta\in ]0,\infty[, $$
whose Laplace transform is given by
$$ \int_0^{\infty}e^{-\lambda \theta}\varpi_{q}(\theta)d\theta=e^{-\lambda^{q}},\quad q\in ]0,1[. $$
(6)
Using the fact (6) again and again, we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equk_HTML.gif
Now we invert the last Laplace transform to get
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equl_HTML.gif
Note that (4) and (5), the proof is completed. □

Remark 3.1

When E=I, I:YY is the identity operator, we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equm_HTML.gif

Definition 3.1

For each https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq10_HTML.gif and ϕC(E), a mild solution of the system (1) we mean a function xC(J,X) which satisfies
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equn_HTML.gif

The following results of https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq12_HTML.gif will be used throughout this paper.

Lemma 3.2

Assume (H1)(H3) hold. The following properties hold:
  1. (i)
    For any fixedt≥0, https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq13_HTML.gifandhttps://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq14_HTML.gifare linear and bounded operators, i.e., for anyxX,
    https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equo_HTML.gif
     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq15_HTML.gifandhttps://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq16_HTML.gifare compact.

     

Proof

(i) For any fixed t≥0, since E−1 and T(t) are linear operator, it is easy to see that https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq17_HTML.gif and https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq18_HTML.gif are also linear operators.

For any xX, according to (4), we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equp_HTML.gif
where we use the fact
$$ \int_0^{\infty}\theta \xi_q(\theta)\,d\theta=\int_0^{\infty}\frac{1}{{\theta}^q} {\varpi}_q(\theta)\,d\theta=\frac{1}{\varGamma(1+q)}. $$
(ii) For each positive constant k, set Yk:={xY:∥x∥≤k}. Then Yk is clearly a bounded subset in Y. We need prove that for any positive constant k and t≥0, the following two sets:
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equr_HTML.gif
are relatively compact in Y. By the assertion (i), we know https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq20_HTML.gif are also linear and bounded. So they maps Yk into a bounded subsets of Y. Then https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq22_HTML.gif are relatively compact in Y for any k>0 and t≥0 due to the compactness of E−1:YX. The proof is complete. □

4 Main Results

In this section, we study the controllability of the system (1) via the well-known Schauder fixed point theorem.

Definition 4.1

The fractional system (1) is said to be controllable on the interval J iff for every continuous initial function ϕC(E) and every x1D(E) there exists a control https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq23_HTML.gif such that the mild solution x of system (1) satisfies x(a)=x1.

In addition to (H1)–(H3), we need the following assumptions:
(H4):
B:UY is a bounded linear operator and a linear operator https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq24_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equs_HTML.gif
has a bounded right inverse operator https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq25_HTML.gif, i.e., WW−1=ID(E), and thus there exist two constants M2, M3>0 such that ∥B∥≤M2 and ∥W−1∥≤M3, where we consider the norm ∥⋅∥D(E) on D(E) for determining M3.
It is obvious that WuD(E) and W is well defined. In fact, it holds
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equt_HTML.gif
for \(q\in ]\frac{1}{2},1[\) and uL2(J,U), meanwhile,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equu_HTML.gif
for q∈]0,1[ and uL(J,U).
We also see that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equ7_HTML.gif
(7)
for any tJ, where \(K_{q}:=\sqrt{\frac{a^{2q-1}}{2q-1}}\) for \(q\in ]\frac{1}{2},1[\) and uL2(J,U), meanwhile, \(K_{q}:=\frac{a^{q}}{q}\) for q∈]0,1[ and uL(J,U).
Next we assume:
(H5):
f satisfies the following two conditions:
  1. (i)

    For each xC the function f(⋅,x):JY is strongly measurable and for each tJ, the function f(t,⋅):CY is continuous.

     
  2. (ii)
    For each k>0, there is a measurable function gk such that
    https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equv_HTML.gif
    for any k>0 sufficiently large and some γ.
     
Note we may take any
$$\gamma>\limsup_{k\to\infty}\frac{a^{q}\|g_k\|_\infty}{qk}. $$
For the sake of simplicity, we state the standard framework to deal with controllability problems as follows. Based on our assumptions, for an arbitrary function x(⋅), it is suitable to define the following control formula:
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equ8_HTML.gif
(8)
In what follows, it is necessary to show that, when using the control u in (8), the operator \(\mathcal {P}\) defined by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equx_HTML.gif
from C(J,X) into C(J,X) for each xC(J,X), has a fixed point. Clearly, this fixed point is just a solution of system (1). Further, one can check
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equy_HTML.gif
which means that u steers the fractional system (1) from ϕ(0) to x1 in finite time a. Consequently, we claim the system (1) is controllable on J.

For each number k>0, define \(\mathcal {B}_{k}:=\{x\in C(J^{*},X): \|x(t)\|\leq k,~t\in J^{*}\}\). Of course, \(\mathcal {B}_{k}\) is clearly a bounded, closed, convex subset in C(J,X).

Under the assumptions (H1)–(H5), we will establish three important results as follows.

Lemma 4.1

There exists a\(K\geq \max\left\{\max_{t\in[-r,0]}\|\phi(t)\|,\frac{M^{*}}{1-\rho}\right\}\)where
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equ9_HTML.gif
(9)
and
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equz_HTML.gif
such that\(\mathcal {P}\mathcal {B}_{K}\subset \mathcal {B}_{K}\).

Proof

Let \(x\in \mathcal {B}_{K}\). If t∈[−r,0] then \(\|(\mathcal {P}x)(t)\|=\|\phi(t)\|\le \max_{t\in[-r,0]}\|\phi(t)\|\). If t∈[0,a] then we derive
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equaa_HTML.gif
where we note that the control u defined in (8) satisfies
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equab_HTML.gif
which implies that
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equ10_HTML.gif
(10)
Thus \(\mathcal {P}\mathcal {B}_{K}\subset \mathcal {B}_{K}\) for any \(K\geq \max\{\max_{t\in[-r,0]}\|\phi(t)\|,\frac{M^{*}}{1-\rho}\}\) sufficiently large. The proof is complete. □

Lemma 4.2

For every fixedtJthe set\(V_{K}(t):=\{(\mathcal {P}x)(t): x\in \mathcal {B}_{K}\} \)is precompact inX.

Proof

This is trivial for t∈[−r,0], since VK(t)={ϕ(t)}. So let 0<t<a be fixed. Note
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equac_HTML.gif
Next, for \(x\in \mathcal {B}_{K}\), we derive
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equad_HTML.gif
and so \(\{(\mathcal {P}_{0}x)(t): x\in \mathcal {B}_{K}\}\) is bounded in Y by (10).

Since E−1:YX is compact, then \((\mathcal {P}x)(t):=E^{-1}\left(\{(\mathcal {P}_{0}x)(t): x\in \mathcal {B}_{K}\}\right)\) is precompact in X. The proof is complete. □

Lemma 4.3

\(\mathcal {P}\mathcal {B}_{K}:= \{\mathcal {P}x : x\in \mathcal {B}_{K}\}\)is equicontinuous.

Proof

Let \(x\in \mathcal {B}_{K}\) and t′,t″∈J such that 0<t′<t″, then
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equae_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equaf_HTML.gif

Note that Lemma 3.2(ii), https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq46_HTML.gif are continuous in the uniform operator topology for t≥0, and supsJ|gK(s)|<∞ and u(⋅) is bounded by (10). We can obtain the terms I1,I3,I5,I6,I7→0 as t″→t′. Moreover, applying Lebesgue’s dominated convergence theorem, one can check the terms I2,I4→0 as t″→t′. Thus, \(\mathcal {P}\mathcal {B}_{K}\) is equicontinuous and also bounded. □

Now we are ready to state the main result in this paper.

Theorem 4.1

Assume (H1)(H5) are satisfied. Then the system (1) is controllable onJprovided that the condition (9) hold.

Proof

It follows Lemmas 4.1–4.3 and the Arzela–Ascoli theorem that \(\mathcal {P}\mathcal {B}_{K}\) is precompact in C(J,X). Hence \(\mathcal {P}\) is a completely continuous operator on C(J,X). From the Schauder fixed point theorem, \(\mathcal {P}\) has a fixed point in \(\mathcal {B}_{K}\). Any fixed point of \(\mathcal {P}\) is a mild solution of the system (1) on J satisfying \((\mathcal {P}x)(t)= x(t)\in X\). Thus, the system (1) is controllable on J. □

5 An Example

Let X=Y=U:=L2[0,π]. Consider the following fractional partial differential equation with control:
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equ11_HTML.gif
(11)
Define A:D(A)⊂XY by Ax:=−xyy and E:D(E)⊂XY by Ex:=xxyy, where each domain D(A), D(E) is given by {xX:x,xy are absolutely continuous, xyyXx(t,0)=x(t,π)=0}. From [28], A and E can be written as
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equag_HTML.gif
respectively, where \(x_{n}(y):= \sqrt{\frac{2}{\pi}}\sin ny,n = 1,2,\ldots\) is the orthonormal set of eigenfunctions of A. Moreover, for any xX we have
$$E^{-1}x=\sum_{n=1}^{\infty}\frac{1}{1+n^{2}}\langle x,x_{n}\rangle x_{n},\quad\quad -AE^{-1}x=\sum_{n=1}^{\infty}\frac{-n^{2}}{1+n^{2}}\langle x,x_{n}\rangle x_{n} $$
and
$$T(t)x=\sum_{n=1}^{\infty}e^{\frac{-n^{2}}{1+n^{2}}t}\langle x,x_{n}\rangle x_{n}. $$
Clearly, E−1 is compact, bounded with ∥E−1∥≤1 and −AE−1 generates the above strongly continuous semigroup T(t) on Y with ∥T(t)∥≤et≤1. Then, the two characterize operators https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq56_HTML.gif can be written as
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equaj_HTML.gif
Clearly,
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equak_HTML.gif
We make the following assumptions:
(C1):
B:UY is defined by B:=bI, b>0 and https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq57_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equal_HTML.gif
Since \(q=\frac{3}{4}>\frac{1}{2}\), we take https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq59_HTML.gif and so \(K_{\frac{3}{4}}:=\sqrt{2}\). Next, let u(s,y):=x(y)∈U. Then
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equam_HTML.gif
where
$$\mathbb{E}_{\frac{3}{4}}\left(-\frac{n^{2}}{1+n^{2}}\right):=\int_{0}^{\infty} e^{-\frac{n^{2}}{1+n^{2}}\theta}\xi_{\frac{3}{4}}(\theta)\,d\theta= \int_{0}^{\infty} e^{-\frac{n^{2}}{1+n^{2}}\theta}\frac{1}{\frac{3}{4}}\theta^{-\big(1+\frac{1}{\frac{3}{4}}\big)}\varpi_{\frac{3}{4}} (\theta^{-\frac{1}{\frac{3}{4}}})\,d\theta, $$
is a Mittag–Leffler function (for more details, see formulas (24)–(27) of [29]).
Note that \(0<1-e^{-\frac{n^{2}}{1+n^{2}}\theta}< 1-e^{-\theta}<1\) for any θ>0. So we have
$$1-\mathbb{E}_{\frac{3}{4}}\left(-\frac{1}{2}\right)\le 1-\mathbb{E}_{\frac{3}{4}}\left(-\frac{n^{2}}{1+n^{2}}\right)\le 1-\mathbb{E}_{\frac{3}{4}}(-1).$$
From the above computations we know that W is surjective. So we define an inverse https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_IEq62_HTML.gif by
$$\bigl(W^{-1}x\bigr)(t,y):=\frac{1}{b}\sum_{n=1}^{\infty}\frac{n^2 \langle x,x_{n}\rangle x_{n}}{\big[1-\mathbb{E}_{\frac{3}{4}} \big(-\frac{n^{2}}{1+n^{2}}\big)\big]}, $$
for \(x=\sum_{n=1}^{\infty}\langle x,x_{n}\rangle x_{n}\). Since
$$\|x\|_{D(E)}:=\|Ex\|:=\sqrt{\sum_{n=1}^{\infty}\bigl(1+n^{2}\bigr)^2\langle x,x_{n}\rangle^2}, $$
for xD(E), we derive
https://static-content.springer.com/image/art%3A10.1007%2Fs10957-012-0174-7/MediaObjects/10957_2012_174_Equar_HTML.gif
Note W−1x is independent of tJ1. Consequently, we obtain
$$\big\|W^{-1}\big\|\le \frac{1}{b\big[1-\mathbb{E}_{\frac{3}{4}}\big(-\frac{1}{2}\big)\big]}. $$
Next, we suppose
(C2):

f:J1×RR. For each xR, f(t,x) is measurable and for each tJ1, f(t,x) is continuous. Moreover, \(\limsup_{k\rightarrow \infty}\frac{1}{k}\sup_{t\in J_{1},|x|\leq k}|f(t,x)|:= \gamma<\infty\).

Define F:J1×C([−1,0],X)→Y by F(t,z)(y)=f(t,z(−r)(y)). Now, the system (11) can be abstracted as
$$\left\{ \begin{array}{l@{\quad}l} ^{C}_{0}D^{\frac{3}{4}}_{t}(Ex(t))=-Ax(t)+F(t,x_{t})+Bu(t),& t\in J_1,\\ [5pt] x(t)=\phi(t),&-1\leq t\leq 0. \end{array} \right. $$
Clearly, all the assumptions in Theorem 4.1 are satisfied, if (9) holds:
$$ \frac{\gamma}{\varGamma(\frac{3}{4})}\left[1+\frac{\sqrt{2}}{\varGamma(\frac{3}{4})\big[1-\mathbb{E}_{\frac{3}{4}}\big(-\frac{1}{2}\big)\big]}\right]<1. $$
(12)
Then the system (11) is controllable on J1.
To compute numerically \(\mathbb{E}_{\frac{3}{4}}(-\frac{1}{2})\), we use the integral representation formula (34) in [30]:
$$\mathbb{E}_{q}(-z):=\frac{\sin(q\pi)}{\pi}\int_{0}^{\infty}\frac{s^{q-1}}{1+2s^{q}\cos(q\pi) +s^{2q}}e^{-z^{\frac{1}{q}}s}\,ds $$
for \(q=\frac{3}{4}\) and \(z=\frac{1}{2}\), to get
$$\mathbb{E}_{\frac{3}{4}}\left(-\frac{1}{2}\right)= \frac{1}{\sqrt{2} \pi}\int_{0}^{\infty}\frac{e^{-\frac{s}{2^{\frac{4}{3}}}}}{ s^{\frac{1}{4}} \big(1-\sqrt{2} s^{\frac{3}{4}}+s^{\frac{3}{2}}\big)}\,ds. $$
Let \(h(s)=1-\sqrt{2} s^{\frac{3}{4}}+s^{\frac{3}{2}}\). It follows h has a unique stationary point \(s^{*}=(\frac{\sqrt{2}}{2})^{\frac{4}{3}}\) and h″(s)>0 for s≥0 that \(h(s)\ge h(s^{*})=\frac{1}{2}\) for s≥0. Thus, we obtain
$$\frac{1}{\sqrt{2} \pi} \int_{\eta}^{\infty}\frac{e^{-\frac{s}{2^{\frac{4}{3}}}}}{ s^{\frac{1}{4}} h(s)}\,ds\le \frac{\sqrt{2}}{\pi \eta^{\frac{1}{4}}}\int_{\eta}^{\infty}e^{-\frac{s}{2^{\frac{4}{3}}}}\,ds=\frac{2^{\frac{4}{3}}\sqrt{2}}{\pi \eta^{\frac{1}{4}}}e^{-\frac{\eta}{2^{\frac{4}{3}}}}\le 0.000004, $$
for η≥30. On the other hand, a numerical computation in Mathematica shows
$$\frac{1}{\sqrt{2} \pi}\int_{0}^{30}\frac{e^{-\frac{s}{2^{\frac{4}{3}}}}}{s^{\frac{1}{4}} h(s)}\,ds\doteq 0.60379. $$
So we get \(\mathbb{E}_{\frac{3}{4}}(-\frac{1}{2})\doteq 0.60379\). Finally, using \(\sqrt{2}\doteq1.41421\), \(1-\mathbb{E}_{\frac{3}{4}}(-\frac{1}{2})\doteq 0.39621\) and \(\varGamma(\frac{3}{4})\doteq 1.22542\), one can find that (12) holds if γ<0.17508.

6 Conclusions

Controllability of fractional functional evolution equations of Sobolev type have been investigated. Utilizing the boundedness and compactness of two new introduced characteristic solution operators with Schauder fixed point theorem, sufficient conditions for controllability of such case of equations corresponding to two certain admissible control sets are provided. Here, we succeed in removing the compactness condition on the semigroup.

Our further work will be devoted to study controllability of the above problems with the help of the theory of propagation family and the techniques of the measure of noncompactness.

Acknowledgements

The authors thank the referees for their careful reading of the manuscript and insightful comments. We also acknowledge the valuable comments and suggestions from the editors. Finally, The first author acknowledges the support by Grants VEGA-MS 1/0507/11, VEGA-SAV 2/0124/12 and APVV-0414-07; the second author acknowledges the support by National Natural Science Foundation of China (11201091) and Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169) and the third author acknowledges the support by National Natural Science Foundation of China (11271309), Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001) and Key Projects of Hunan Provincial Natural Science Foundation of China (12JJ2001).

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